<s>
In	O
model	B-Application
checking	I-Application
,	O
a	O
field	O
of	O
computer	B-General_Concept
science	I-General_Concept
,	O
a	O
region	O
is	O
a	O
convex	O
polytope	O
in	O
for	O
some	O
dimension	O
,	O
and	O
more	O
precisely	O
a	O
zone	O
,	O
satisfying	O
some	O
minimality	O
property	O
.	O
</s>
<s>
Furthermore	O
,	O
when	O
those	O
vectors	O
are	O
considered	O
as	O
a	O
tuple	O
of	O
clocks	B-Application
,	O
both	O
vectors	O
have	O
the	O
same	O
set	O
of	O
possible	O
futures	O
.	O
</s>
<s>
Intuitively	O
,	O
it	O
means	O
that	O
any	O
timed	O
propositional	O
temporal	O
logic-formula	O
,	O
or	O
timed	B-Application
automaton	I-Application
or	O
signal	B-Application
automaton	I-Application
using	O
only	O
the	O
constraints	O
of	O
can	O
not	O
distinguish	O
both	O
vectors	O
.	O
</s>
<s>
Taking	O
a	O
product	O
of	O
this	O
region	O
automaton	O
and	O
of	O
a	O
timed	B-Application
automaton	I-Application
which	O
accepts	O
a	O
language	O
creates	O
a	O
finite	B-Architecture
automaton	I-Architecture
or	O
a	O
Büchi	B-General_Concept
automaton	I-General_Concept
which	O
accepts	O
untimed	O
.	O
</s>
<s>
In	O
particular	O
,	O
it	O
allows	O
to	O
reduce	O
the	O
emptiness	O
problem	O
for	O
to	O
the	O
emptiness	O
problem	O
for	O
a	O
finite	O
or	O
Büchi	B-General_Concept
automaton	I-General_Concept
.	O
</s>
<s>
This	O
technique	O
is	O
used	O
for	O
example	O
by	O
the	O
software	O
UPPAAL	B-Application
.	O
</s>
<s>
Let	O
a	O
set	O
of	O
clocks	B-Application
.	O
</s>
<s>
The	O
definition	O
of	O
a	O
region	O
over	O
the	O
clocks	B-Application
of	O
uses	O
those	O
numbers	O
'	O
s	O
.	O
</s>
<s>
Two	O
clocks	B-Application
assignments	O
and	O
are	O
equivalent	O
if	O
they	O
satisfy	O
the	O
following	O
constraints	O
:	O
</s>
<s>
furthermore	O
,	O
for	O
each	O
pair	O
of	O
clocks	B-Application
,	O
where	O
contains	O
constraints	O
of	O
the	O
form	O
and	O
,	O
then	O
contains	O
an	O
(	O
in	O
)	O
equality	O
of	O
the	O
form	O
with	O
being	O
either	O
=	O
,	O
<	O
or	O
≤	O
.	O
</s>
<s>
It	O
suffices	O
,	O
for	O
each	O
clock	O
,	O
to	O
state	O
to	O
which	O
interval	O
it	O
belongs	O
and	O
to	O
recall	O
the	O
order	O
of	O
the	O
fractional	O
part	O
of	O
the	O
clocks	B-Application
which	O
belong	O
in	O
an	O
open	O
interval	O
of	O
length	O
1	O
.	O
</s>
<s>
It	O
follows	O
that	O
the	O
size	O
of	O
this	O
structure	O
is	O
with	O
the	O
number	O
of	O
clocks	B-Application
.	O
</s>
<s>
While	O
this	O
definition	O
is	O
more	O
abstract	O
,	O
it	O
is	O
also	O
the	O
reason	O
why	O
regions	O
are	O
used	O
in	O
model	B-Application
checking	I-Application
.	O
</s>
<s>
Intuitively	O
,	O
this	O
definition	O
states	O
that	O
two	O
clock	O
assignments	O
belong	O
to	O
the	O
same	O
region	O
if	O
the	O
differences	O
between	O
them	O
are	O
such	O
that	O
no	O
timed	B-Application
automaton	I-Application
can	O
notice	O
them	O
.	O
</s>
<s>
Given	O
a	O
set	O
of	O
clock	O
,	O
two	O
assignments	O
two	O
clocks	B-Application
assignments	O
and	O
belongs	O
to	O
the	O
same	O
region	O
if	O
for	O
each	O
timed	B-Application
automaton	I-Application
in	O
which	O
the	O
guards	O
never	O
compare	O
a	O
clock	O
to	O
a	O
number	O
greater	O
than	O
,	O
given	O
any	O
location	O
of	O
,	O
there	O
is	O
a	O
timed	O
bisimulation	B-Application
between	O
the	O
extended	O
states	O
and	O
.	O
</s>
<s>
More	O
precisely	O
,	O
this	O
bisimulation	B-Application
preserves	O
letters	O
and	O
locations	O
but	O
not	O
the	O
exact	O
clock	O
assignments	O
.	O
</s>
<s>
Given	O
a	O
region	O
defined	O
by	O
a	O
set	O
of	O
(	O
in	O
)	O
equations	O
,	O
and	O
a	O
set	O
of	O
clocks	B-Application
,	O
the	O
region	O
similar	O
to	O
in	O
which	O
the	O
clocks	B-Application
of	O
are	O
restarted	O
is	O
now	O
defined	O
.	O
</s>
<s>
Let	O
the	O
set	O
of	O
clocks	B-Application
such	O
that	O
contains	O
the	O
constraint	O
.	O
</s>
<s>
Let	O
the	O
set	O
of	O
clocks	B-Application
such	O
that	O
there	O
are	O
no	O
constraints	O
of	O
the	O
form	O
in	O
.	O
</s>
<s>
the	O
constraints	O
of	O
not	O
using	O
the	O
clocks	B-Application
of	O
,	O
</s>
<s>
There	O
are	O
at	O
most	O
regions	O
,	O
where	O
is	O
the	O
number	O
of	O
clocks	B-Application
.	O
</s>
<s>
Given	O
a	O
timed	B-Application
automaton	I-Application
,	O
its	O
region	O
automaton	O
is	O
a	O
finite	B-Architecture
automaton	I-Architecture
or	O
a	O
Büchi	B-General_Concept
automaton	I-General_Concept
which	O
accepts	O
untimed	O
.	O
</s>
<s>
This	O
automaton	O
is	O
similar	O
to	O
,	O
where	O
clocks	B-Application
are	O
replaced	O
by	O
region	O
.	O
</s>
<s>
Let	O
a	O
timed	B-Application
automaton	I-Application
.	O
</s>
<s>
The	O
region	O
automaton	O
of	O
,	O
denoted	O
by	O
is	O
a	O
finite	O
or	O
Büchi	B-General_Concept
automaton	I-General_Concept
which	O
is	O
essentially	O
a	O
product	O
of	O
and	O
of	O
the	O
region	O
graph	O
defined	O
above	O
.	O
</s>
<s>
Since	O
two	O
clocks	B-Application
assignment	O
belonging	O
to	O
the	O
same	O
region	O
satisfies	O
the	O
same	O
guard	O
,	O
each	O
region	O
contains	O
enough	O
information	O
to	O
decide	O
which	O
transitions	O
can	O
be	O
taken	O
.	O
</s>
<s>
its	O
set	O
of	O
accepting	B-Architecture
states	I-Architecture
is	O
,	O
</s>
